If-f-x-f-x-1-x-1-x-find-f-2-

Question Number 118513 by I want to learn more last updated on 18/Oct/20

If f (x) + f (\frac{x - 1}{x}) = 1 + x, find f (2) .

Answered by Dwaipayan Shikari last updated on 18/Oct/20

f (2) + f (\frac{2 - 1}{2}) = 1 + 2 \to (a) (T a k n g x = 2)

f (\frac{1}{2}) + f (\frac{\frac{1}{2} - 1}{\frac{1}{2}}) = 1 + \frac{1}{2} \to (b) (T a k i n g x = \frac{1}{2})

f (2) - f (- 1) = \frac{3}{2} \to (a) - (b)

f (- 1) + f (\frac{- 1 - 1}{- 1}) = 1 - 1 \to (f) (T a k i n g x = - 1)

2 f (2) = \frac{3}{2} \to (a) - (b) + (f)

f (2) = \frac{3}{4}

Commented by I want to learn more last updated on 18/Oct/20

Thanks sir, but please i {don}^{'} t understand the changement of nunbers .

like the second line and the like .

Commented by I want to learn more last updated on 18/Oct/20

Thanks sir, i understand now .

Answered by benjo_mathlover last updated on 18/Oct/20

G i v e n (1) f (x) + f (\frac{x - 1}{x}) = 1 + x

r e p l a c i n g x b y \frac{x - 1}{x}

(2) f (\frac{x - 1}{x}) + f (\frac{\frac{x - 1}{x} - 1}{\frac{x - 1}{x}}) = 1 + \frac{x - 1}{x}

\Rightarrow f (\frac{x - 1}{x}) + f (\frac{1}{1 - x}) = \frac{2 x - 1}{x}

r e p l a c i n g \frac{x - 1}{x} b y x

(3) f (\frac{1}{1 - x}) + f (x) = 1 + \frac{1}{1 - x}

\Rightarrow f (\frac{1}{1 - x}) + f (x) = \frac{2 - x}{x}

a d d i n g (1), (2) a n d (3); g i v e s

2 [f (x) + f (\frac{x - 1}{x}) + f (\frac{1}{1 - x})] = 1 + x + \frac{2 x - 1}{x} + \frac{2 - x}{x}

s u b s t i t u t e e q u a t i o n (2)

2 [f (x) + \frac{2 x - 1}{x}] = \frac{x^{2} + 2 x + 1}{x}

f (x) = \frac{x^{2} + 2 x + 1}{2 x} - \frac{2 x - 1}{x}

f (x) = \frac{x^{2} - 2 x + 3}{2 x} \Rightarrow f (2) = \frac{4 - 4 + 3}{4}

\Rightarrow f (x) = \frac{3}{4}

Commented by I want to learn more last updated on 18/Oct/20

Thanks sir .